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1

Both equations for $S, v$ should remain the same as they govern the evolution of these quantities over time regardless of initial conditions. It is the initial condition (unstated here) that must change: $v_0 \rightarrow v_0 + \epsilon$.


0

The historical volatilities of the market factors is not the same as the implied volatilty used to price the options. The "implied volatility" is just one of the model inputs. It does not need to be similar to the historical volatility of the underlying. The mark to market of an option is the premium that one would have to pay in the market for this option. ...


3

The simplest long vol strategy is to be long an ATM straddle and delta hedge it, the problem is that when it is no longer ATM the exposure to vol weakens. You could then sell that straddle and enter another ATM one. Another solution is the vol swap or variance swap mentioned by Stephane below. It gives constant exposure no matter what the level of S&P. ...


2

I do not mean to discourage you, but it sounds like you're a wee bit late for this round of volatility games, for two reasons: You are still trying to figure out how to implement a long vol strategy. The market has already priced the risk in, i.e. buying volatility is already expensive. However, never too late to learn and prepare for a next time. My ...


3

What not to do What you are asking us, without knowing, is related to how to price a variance swap. Well, under a general diffusion process, variance swaps can be priced by forming a suitably weighted portfolio of options over a continuum of strike prices with the entire portfolio maturing on a given date. The intuition is that your exposure to volatility ...


3

Suppose that you are riskless asset with return $r_{ft}$ and a risky asset with return $r_t$ and conditional volatility $\sigma_t(r_t) := \sqrt{V_t(r_t)}$. We build a portfolio using weights $(w_1, w_2) \in \mathbb{R}$, or as you wrote it $w_t := w_{1t}$, $w_{2t} := 1 - w_t$. This portfolio will have a time $t$ return of $r_{pt}$. Its volatility is given by $...


3

Let me venture a guess. If I had to design a system from scratch, I would probably prefer GARCH processes to properly stochastic conditional volatility processes. The fact that one step ahead, the conditional volatility process is known makes filtering both trivial and faster. Moreover, this class of option pricing model affords me all the flexibility of ...


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